Graphics Reference
In-Depth Information
In qns 83 and 84 we have established that a convergent sequence
definition of limit from above and a neighbourhood definition of limit
from above are equivalent.
85 State a neighbourhood definition of limit from below equivalent to
the convergent sequence definition of limit from below given after
qn 73.
Two-sided limits
De
fi
nition of continuitybylimits
l
and lim
l
If both lim
f
(
x
)
f
(
x
)
it is customary to write
lim
f
(
x
)
l
or 'as
x
a
,
f
(
x
)
l
' and to say 'as
x
tends to
a
,
f
(
x
) tends to
l
'.
86 If
f
: R
R is continuous at
a
, useqns 75 and 78 to show that
lim
f
(
x
)
f
(
a
).
87 Write down the neighbourhood definitions of
l
and lim
l
.
lim
f
(
x
)
f
(
x
)
Use these definitions to show that, if lim
f
(
x
)
l
,
then, given
such that when
x
is in thedomain
of
f
and 0
x
a
, then
f
(
x
)
l
.
0, there exists a
The important point here is recognising that 0
x
a
means
either a
x
aora
x
a
and theanswr to thequstion consists of showing how to choose
the
.
88
(i) If, given
0, there exists a
such that
f
(
x
)
l
when
0
x
a
, provethat
l
.
lim
f
(
x
)
Use neighbourhoods for this proof, not sequences.
